The theory of electron heating in collisonless plasma shock waves

Abstract

Equations are derived to describe the evolution of an electron distribution function under the action of electromagnetic instabilities in a non-uniform plasma using an extension of the quasilinear theory of Kennel and Engelmann. Variations in both the electron density and temperature and the background magnetic field are taken into account. These equations are simplified in the limit of small electron beta so that an electrostatic approximation is justified. Methods are then presented which allow the solution of these equations (or, in principle, the more complex electromagnetic equations). In particular, a method of solving the kinetic dispersion relation for an arbitrary background (first-order) distribution function with the minimum of additional assumptions and approximations is described in detail. The electrostatic equations are solved for a number of different cases in order to study the action of the modified two stream instability on the electron distribution function. Throughout, realistic values of the ratios of electron to ion mass and electron plasma to cyclotron frequency ratio are used. The applications to collisionless plasma shock waves are discussed, and it is found that the modified two stream instability can produce the (relatively small) amounts of electron heating observed at quasi-perpendicular terrestrial bow shocks, and the flat-topped electron distribution functions seen to evolve. Extensions to the model which would greatly improve its applicability and accuracy, as well as the amount of computational effort required, are discussed.